#140718

order	512
nilpotency class	3
isoclinism	isoclinic class of #1705
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1954
prime radical	#3169	C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

356 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

(2,3)(4,5)(7,8)(9,10,14,19,15,18,21,34,13,17,28,33,12,26,27,30)(11,20,24,36,25,31,37,40,23,35,38,39,22,16,29,32);(1,2,5,7,6,8,4,3)(9,11,21,37,12,22,14,24,13,23,27,29,15,25,28,38)(10,16,30,39,26,35,33,40,17,31,34,36,18,20,19,32);(1,3,4,8,6,7,5,2)(10,17)(16,31)(18,26)(19,33)(20,35)(30,34)(32,40)(36,39)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=eg,d^2=g,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47914);